a solenoid with an iron core and a bulb are connected to an ac source. how does the brightness of the bulb change when the iron core is removed from the solenoid?

yes the brightness of the bulb increases when we take out the iron core from the solenoid.
this is because, when the iron core is taken out the inductance decreases as the inductance decreases the capacitive reactance also decreases hence the current flowing increases which makes the bulb glow brighter.